My smoke is not animating at all
The problem is the adding of the keyframe - what it appears that you are doing is moving the mouse up into the 3D View and pressing I. This is adding a keyframe to the selected object in the scene. What you should actually be doing is hovering the mouse over the rotation value in the Mapping node (the value you just changed) and pressing I - this will set the keyframe on the node value (and make it turn yellow) so that it becomes animated.